This solution applies to the following achievements:
Shortly after you start Angel Island Zone Act 1, you will find a giant ring which will take you to a bonus stage. You can very easily get 50,000 points here simply by collecting every ring in the special stage. That can be a little tricky, especially if it's your first time playing the game but a couple of tries and you should have it. You'll also get a shiny chaos emerald at the end of the stage which will get you this achievement:
And contribute to this one:
In the special stage you will find lots of blue and red spheres. You complete the stage by turning every blue sphere red (just by running through them) and you fail the stage if you hit a red. The way you get the rings is by looking for any group of spheres arranged in a square bigger than 2x2. What you want to do is change the spheres on the outside
of the square red so that there are still some blue squares enclosed in the reds. As soon as all the spheres on the outline have been changed all of them will change to rings, simply go back and collect them. Collecting every ring available in the stage gets you a perfect bonus worth 50,000. To get the Points III achievement, which requires 80,000 points, simply repeat the process by finding another giant ring and collecting every ring or continue playing the game as normal because you'll be able to get another 30,000 points no problem within a few acts. Although you will need to find six other giant rings anyway in order to get all chaos emeralds for the achievements mentioned above.
Hope this helps :)